Detecting and modeling nonlinearity in the gas furnace data

Using a modification of the Hinich, J Time Ser Anal 3(3):169–176, (1982) bispectrum test for nonlinearity and Gaussianity, the residuals of the Tiao and Box, J Am Stat Assoc 76:802–816, (1981) constrained and unconstrained VAR models for the gas furnace data reject the assumption of Gaussianity and linearity over a grid of bandwidths for estimating the bispectrum. These findings call into question the specification of the linear VAR and VARMA models assumed by Tiao and Box, J Am Stat Assoc 76:802–816, (1981). Utilizing the alternative Hinich J Nonparametr Stat 6:205–221, (1996) nonlinearity test, the residuals of the VAR model were shown to exhibit episodic nonlinearity. The sensitivity of the findings to outliers is investigated by estimating and testing the residuals of L1 and MINIMAX models from 1–6 lags. Building on the linear dynamic specification, a multivariate adaptive regression splines (MARS) model is estimated, using two software implementations, and shown to remove the nonlinearity in the residuals. Leverage plots were used to illustrate the “cost” of imposing a linearity assumption. Out-of-sample forecasting tests from 1–6 periods ahead found that using the sum-of-squared errors criteria, the MARS model out performed ACE, GAM and projection pursuit models.     

Visualizing hypothesis tests in multivariate linear models: the heplots package for R

Hypothesis-error (or “HE”) plots, introduced by Friendly (J Stat Softw 17(6):1–42, 2006a; J Comput Graph Stat 16:421–444, 2006b), permit the visualization of hypothesis tests in multivariate linear models by representing hypothesis and error matrices of sums of squares and cross-products as ellipses. This paper describes the implementation of these methods in the heplots package for R, as well as their extension, for example from two to three dimensions and by scaling hypothesis ellipses and ellipsoids in a natural manner relative to error. This is a paper for the proceedings of the Directions in Statistical Computing conference.     

Combinatorial Groupoids,Cubical Complexes,and the Lovász Conjecture

This paper lays the foundation for a theory of combinatorial groupoids that allows us to use concepts like “holonomy”, “parallel transport”, “bundles”, “combinatorial curvature”, etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes and other combinatorial objects. We introduce a new, holonomy-type invariant for cubical complexes, leading to a combinatorial “Theorema Egregium” for cubical complexes that are non-embeddable into cubical lattices. Parallel transport of Hom-complexes and maps is used as a tool to extend Babson–Kozlov–Lovász graph coloring results to more general statements about nondegenerate maps (colorings) of simplicial complexes and graphs. The author was supported by grants 144014 and 144026 of the Serbian Ministry of Science and Technology.     

On trees covering chains or stars

In this paper, in the context of the “dessins d’enfants” theory, we give a combinatorial criterion for a plane tree to cover a tree from the classes of “chains” or “stars.” We also discuss some applications of this result that are related to the arithmetical theory of torsion on curves. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 6, pp. 207–215, 2007.     

A note on the prior parameter choice in finite mixture models of distributions from exponential families

This work presents a new scheme to obtain the prior distribution parameters in the framework of Rufo et al. (Comput Stat 21:621–637, 2006). Firstly, an analytical expression of the proposed Kullback–Leibler divergence is derived for each distribution in the considered family. Therefore, no previous simulation technique is needed to estimate integrals and thus, the error related to this procedure is avoided. Secondly, a global optimization algorithm based on interval arithmetic is applied to obtain the prior parameters from the derived expression. The main advantage by using this approach is that all solutions are found and rightly bounded. Finally, an application comparing this strategy with the previous one illustrates the proposal.     

Measurable Riemannian geometry on the Sierpinski gasket: the Kusuoka measure and the Gaussian heat kernel estimate

We study the standard Dirichlet form and its energy measure,called the Kusuoka measure, on the Sierpinski gasket as aprototype of “measurable Riemannian geometry”. The shortest pathmetric on the harmonic Sierpinski gasket is shown to be thegeodesic distance associated with the “measurable Riemannianstructure”. The Kusuoka measure is shown to have the volumedoubling property with respect to the Euclidean distance and alsoto the geodesic distance. Li–Yau type Gaussian off-diagonal heatkernel estimate is established for the heat kernel associated withthe Kusuoka measure.     

Model selection bias and Freedman’s paradox

In situations where limited knowledge of a system exists and the ratio of data points to variables is small, variable selection methods can often be misleading. Freedman (Am Stat 37:152–155, 1983) demonstrated how common it is to select completely unrelated variables as highly “significant” when the number of data points is similar in magnitude to the number of variables. A new type of model averaging estimator based on model selection with Akaike’s AIC is used with linear regression to investigate the problems of likely inclusion of spurious effects and model selection bias, the bias introduced while using the data to select a single seemingly “best” model from a (often large) set of models employing many predictor variables. The new model averaging estimator helps reduce these problems and provides confidence interval coverage at the nominal level while traditional stepwise selection has poor inferential properties.     

Acceleration of the EM algorithm using the vector epsilon algorithm

The expectation–maximization (EM) algorithm is a very general and popular iterative computational algorithm to find maximum likelihood estimates from incomplete data and broadly used to statistical analysis with missing data, because of its stability, flexibility and simplicity. However, it is often criticized that the convergence of the EM algorithm is slow. The various algorithms to accelerate the convergence of the EM algorithm have been proposed. The vector ε algorithm of Wynn (Math Comp 16:301–322, 1962) is used to accelerate the convergence of the EM algorithm in Kuroda and Sakakihara (Comput Stat Data Anal 51:1549–1561, 2006). In this paper, we provide the theoretical evaluation of the convergence of the ε-accelerated EM algorithm. The ε-accelerated EM algorithm does not use the information matrix but only uses the sequence of estimates obtained from iterations of the EM algorithm, and thus it keeps the flexibility and simplicity of the EM algorithm.     

A filter inexact-restoration method for nonlinear programming

A new iterative algorithm based on the inexact-restoration (IR) approach combined with the filter strategy to solve nonlinear constrained optimization problems is presented. The high level algorithm is suggested by Gonzaga et al. (SIAM J. Optim. 14:646–669, 2003) but not yet implement—the internal algorithms are not proposed. The filter, a new concept introduced by Fletcher and Leyffer (Math. Program. Ser. A 91:239–269, 2002), replaces the merit function avoiding the penalty parameter estimation and the difficulties related to the nondifferentiability. In the IR approach two independent phases are performed in each iteration, the feasibility and the optimality phases. The line search filter is combined with the first one phase to generate a “more feasible” point, and then it is used in the optimality phase to reach an “optimal” point. Numerical experiences with a collection of AMPL problems and a performance comparison with IPOPT are provided.      

On the reduction to a complete class in multiple decision problems (2)

Summary As a criterion for the reduction to a complete class of decision rule in case where actions, samples and states are finite in number, “regret-relief ratio” criterion and “incremental loss-gain ratio” criterion were introduced in 2-state of nature case [2]. In this paper, “generalized regret-relief ratio” criterion ink-state of nature case is introduced as an extension of “regret-relief ratio” criterion and its usefulness is shown with an example. The Institute of Statistical Mathematics     

The divergence of Banach space valued random variables on Wiener space

The domain of definition of the divergence operator δ on an abstract Wiener space (W,H,μ) is extended to include W–valued and – valued “integrands”. The main properties and characterizations of this extension are derived and it is shown that in some sense the added elements in δ’s extended domain have divergence zero. These results are then applied to the analysis of quasiinvariant flows induced by W-valued vector fields and, among other results, it turns out that these divergence-free vector fields “are responsible” for generating measure preserving flows. Mathematics Subject Classification (2000): Primary 60H07, Secondary 60H05 An erratum to this article is available at .     

Reduction of hugoniot-maslov chains for trajectories of solitary vortices of the “shallow water” equations to the hill equation

According to Maslov’s idea, many two-dimensional, quasilinear hyperbolic systems of partial differential equations admit only three types of singularities that are in general position and have the property of “structure self-similarity and stability.” Those are: shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is described by an infinite chain of ordinary differential equations (ODE) that generalize the well-known Hugoniot conditions for shock waves. After some reasonable closure of the chain for the case of solitary vortices in the “shallow water” equations, we obtain a nonlinear system of sixteen ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential. This means that, in some approximations, the trajectory of a solitary vortex can be described by the Hill equation. This result can be used to predict the trajectory of the vortex center if we know its observable part. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 47–66.     

The Mathematicians’ Happy Hunting Ground: Einstein’s General Theory of Relativity

There is hardly any doubt that for physics special relativity theory is of much greater consequence than the general theory. The reverse situation prevails with respect to mathematics: there special relativity theory had comparatively little, general relativity theory very considerable, influence, above all upon the development of a general scheme for differential geometry. —Hermann Weyl, “Relativity as a Stimulus to Mathematical Research,” pp. 536–537.     

Polynomiality criterion for entire functions of several complex variables

The “radial” polynomiality criterion for entire functions of several complex variables is proved. Translated fromMatermaticheskie Zametki, Vol. 66, No. 4, pp. 500–502, October, 1999.     

Monotonicity in asymmetric first-price auctions with affiliation

I study monotonicity of equilibrium strategies in first-price auctions with asymmetric bidders, risk aversion, affiliated types, and interdependent values. Every mixed-strategy equilibrium is shown to be outcome-equivalent to a monotone pure-strategy equilibrium under the “priority rule” for breaking ties. This provides a missing link to establish uniqueness in the “general symmetric model” of Milgrom and Weber (Econometrica 50:1089–1122, 1982). Non-monotone equilibria can exist under the “coin-flip rule” but they are distinguishable: all non-monotone equilibria have positive probability of ties whereas all monotone equilibria have zero probability of ties. This provides a justification for the standard empirical practice of restricting attention to monotone strategies. Hendricks et al. (2003) provide an overview of recent empirical work. For a survey of experimental work, see Kagel and Levin (2002).     

Hilbert–Kunz multiplicity, McKay correspondence and good ideals¶in two-dimensional rational singularities

Hilbert–Kunz multiplicity is known to be a very mysterious invariant of a ring or an ideal. We will show a very beautiful formula on Hilbert–Kunz multiplicity for integrally closed ideals in two-dimensional Gorenstein rational singularities. In the proof, “McKay correspondence” and “Riemann–Roch formula” play essential roles. Also this formula gives a new significance to “good ideals”. Received: 25 October 2000     

The ideal of separants in the ring of differential polynomials

We obtained the criterion of existence of a quasi-linear polynomial in a differential ideal in the ordinary ring of differential polynomials over a field of characteristic zero. We generalized the “going up” and “going down” theorems onto the case of Ritt algebras. In particular, new finiteness criteria for differential standard bases and estimates that characterize calculation complexity were obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 215–227, 2007.     

The number of rooted Eulerian planar maps

In this paper we provide a solution of the functional equation unsolved in the paper, by the second author, "On functional equations arising from map enumerations" that appeared in Discrete Math, 123: 93-109 (1993). It is also the number of combinatorial distinct rooted general eulerian planar maps with the valency of root-vertex, the number of non-root vertices and non-root faces of the maps as three parameters. In particular, a result in the paper, by the same author, "On the number of eulerian planar map...     

A Discrete Laplace–Beltrami Operator for Simplicial Surfaces

We define a discrete Laplace–Beltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called “cotan formula”) except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete Laplace–Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa’s Theorem we show that, as claimed, Musin’s harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces. Research for this article was supported by the DFG Research Unit 565 “Polyhedral Surfaces” and the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.     

The Gibbs Cloner for Combinatorial Optimization, Counting and Sampling

We present a randomized algorithm, called the cloning algorithm, for approximating the solutions of quite general NP-hard combinatorial optimization problems, counting, rare-event estimation and uniform sampling on complex regions. Similar to the algorithms of Diaconis–Holmes–Ross and Botev–Kroese the cloning algorithm is based on the MCMC (Gibbs) sampler equipped with an importance sampling pdf and, as usual for randomized algorithms, it uses a sequential sampling plan to decompose a “difficult” problem into a sequence of “easy” ones. The cloning algorithm combines the best features of the Diaconis–Holmes–Ross and the Botev–Kroese. In addition to some other enhancements, it has a special mechanism, called the “cloning” device, which makes the cloning algorithm, also called the Gibbs cloner fast and accurate. We believe that it is the fastest and the most accurate randomized algorithm for counting known so far. In addition it is well suited for solving problems associated with the Boltzmann distribution, like estimating the partition functions in an Ising model. We also present a combined version of the cloning and cross-entropy (CE) algorithms. We prove the polynomial complexity of a particular version of the Gibbs cloner for counting. We finally present efficient numerical results with the Gibbs cloner and the combined version, while solving quite general integer and combinatorial optimization problems as well as counting ones, like SAT.